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Converting
Any String State to Traveling Slope-Wave Components
We verified in §G.3.1 above that traveling-wave components
and in Eq. (G.14) satisfy the ideal string wave equation
. By definition, the physical string displacement is
given by the sum of the traveling-wave components, or
|
(G.15) |
Thus, given any pair of traveling waves and , we can
compute a corresponding string displacement . This leads to the
question whether any initial string state can be converted to a pair
of equivalent traveling-wave components. If so, then d'Alembert's
traveling-wave solution is complete, and all solutions to the
ideal string wave equation can be expressed in terms of traveling waves.
The state of an ideal string at
time is classically specified by its displacement and
velocity
for all [297]. Equation (G.15) gives us as a simple sum of
the traveling-wave components, and now we need a formula for in
terms of them as well. It will be derived in §G.7.3
(see Equations (G.43-G.45)) that we can write
where denotes the partial derivative with respect to as
usual. We thus have
Inverting the two-by-two differential operator matrix yields left- and
right-going slope waves as a function of an arbitrary initial
slope and velocity:
Integrating both sides with respect to , and choosing the constant
of integration to give the correct constant component of , we
obtain the displacement-wave components
where
Notice that if the initial velocity is zero, each of the initial
traveling displacement waves is simply half the initial displacement,
as expected. On the other hand, if the initial displacement is zero
and there is a uniform initial velocity (the whole string is moving),
the initial displacement-wave components are unbounded as the
string length goes to infinity. Related discussion appears in
Appendix M.
It will be seen in §G.7.4 that state conversion between
physical variables and traveling-wave components is more natural and
straightforward when force and velocity are chosen as
the physical state variables (as opposed to displacement and velocity
used here).
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